comprehensive computational model of single- and dual-loop...

Comprehensive computational model ofsingle- and dual-loop optoelectronic

oscillators with experimental verification

Etgar C. Levy,1,∗ Olukayode Okusaga,3 Moshe Horowitz,1 CurtisR. Menyuk,2 Weimin Zhou,3 and Gary M. Carter2

1Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa 32000Israel

2Department of Electrical Engineering, University of Maryland, Baltimore, Maryland 21250USA

3U.S. Army Research Laboratory, Adelphi, Maryland USA∗[email protected]

Abstract: We describe a comprehensive computational model for single-loop and dual-loop optoelectronic oscillators (OEOs). The model takesinto account the dynamical effects and noise sources that are required toaccurately model OEOs. By comparing the computational and experimentalresults in a single-loop OEO, we determined the amplitudes of the whitenoise and flicker noise sources. We found that the flicker noise sourcecontains a strong component that linearly depends on the loop length.Therefore, the flicker noise limits the performance of long-cavity OEOs(� 5 km) at low frequencies ( f < 500 Hz). The model for a single-loop OEOwas extended to model the dual-loop injection-locked OEO (DIL-OEO).The model gives the phase-noise, the spur level, and the locking range ofeach of the coupled loops in the OEO. An excellent agreement between the-ory and experiment is obtained for the DIL-OEO. Due to its generality andaccuracy, the model is important for both designing OEOs and studying thephysical effects that limit their performance. We demonstrate theoreticallythat it is possible to reduce the first spur in the DIL-OEO by more than 20dB relative to its original performance by changing its parameters. Thistheoretical result has been experimentally verified.

© 2010 Optical Society of America

OCIS codes: (060.2320) Fiber optics amplifiers and oscillators; (230.0250) Optoelectronics;(230.4910) Oscillators.

References and links1. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A 13, 1725–1735 (1996).2. W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-

low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005).3. D. Dahan, E. Shumakher, and G. Eisenstein, “Self-starting ultralow-jitter pulse source based on coupled opto-

electronic oscillators with an intracavity fiber parametric amplifier,” Opt. Lett. 30, 1623–1625 (2005).4. C. R. Menyuk, E. C. Levy, O. Okusaga, G. M. Carter, M. Horowitz, and W. Zhou “An analytical model of the

dual-injection-locked opto-electronic oscillator,” IFCS (2009).5. Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola and P. Colet, “Dynamic instabilities of mi-

crowaves generated with optoelectronic oscillators,” Opt. Lett. 32, 2571–2573 (2007).

#131695 - $15.00 USD Received 14 Jul 2010; revised 3 Sep 2010; accepted 7 Sep 2010; published 24 Sep 2010(C) 2010 OSA 27 September 2010 / Vol. 18, No. 20 / OPTICS EXPRESS 21461

6. E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling opto-electronic oscillators,” J. Opt. Soc. Am. B 26,148–159 (2009).

7. O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulationstudy of dual injection-locked OEOs,” IFCS (2009).

8. O. Okusaga, E. J. Adles, E. C. Levy, M. Horowitz, G. M. Carter, C. R. Menyuk, and W. Zhou, “Spurious modesuppression in dual injection-locked optoelectronic oscillators,” IFCS (2010).

9. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optolec-tronic oscillators,” IFCS (2009).

10. E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurementof microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005).

11. N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/ f α power law noise genera-tion,” IEEE Proc. 83, 802–827 (1995).

1. Introduction

Optoelectronic oscillators (OEOs), first introduced by Yao and Maleki in 1996 [1], are usedto generate signals in X-band with a low phase noise. These hybrid opto-electronic devicescontain a long optical fiber, typically in the range of 4–6 km. The RF signal is modulated ontoan optical carrier at the entry to the fiber, which is then demodulated at the exit. Thus, theOEO is effectively a high-Q RF cavity. Such a high-Q cavity makes possible the generation ofa high-frequency signal whose phase noise is nearly independent of the oscillation frequency.Furthermore, the use of a long optical fiber and a tunable narrowband RF filter makes it possibleto tune the oscillating signal over a very broad frequency range. These important advantagesmake the OEO an attractive candidate to replace classical oscillators such as multiplied quartzcrystals or phase-locked dielectric resonator oscillators.

Due to the length of the optical fiber, the cavity mode spacing is too small to filter out asingle cavity mode using an RF filter. Therefore, the RF spectrum of single-loop OEOs containsstrong spurs at the cavity mode frequencies. OEOs with two or more coupled cavities havebeen used to reduce the spurs [2, 3]. In the work of Zhou et al. [2], a long loop OEO, calledthe master loop, generates an RF signal with a low phase noise. The spurs of the master loopare significantly attenuated by injection-locking the master loop to another short-loop OEO,called the slave loop. The OEO has several parameters that can be varied over a wide range,and determining the performance at frequencies close-in to the the carrier with even one set ofparameters is time-consuming. As a consequence, it is not possible to comprehensively explorethe parameter space experimentally; so it is essential to develop a comprehensive model that iscapable of accurately exploring the OEO performance as a function of the OEO parameters.

Several models have been presented for studying the phase noise in OEOs. Yao and Malekihave presented an analytical expression for the phase noise in a single-loop OEO [1] using amodel that assumes that the signal’s change per round-trip at any point in the loop is small andtreats all the loop elements as distributed, rather than lumped. The Yao-Maleki model gives thedependance of the time-averaged phase noise on the frequency offset from the carrier frequency,the cavity length, the oscillation power, and the amplitude of the white noise source. In [4] weintroduced a reduced model for calculating the phase noise in a dual-injection-locked OEO(DIL-OEO) that is based on the Yao and Maleki model, and like that model assumes that thesignal’s change per round-trip at any point in both loops is small. Chembo et al. [5] developeda model for a single-loop OEO that is based on a delay-differential equation for studying thesignal dynamics in single-loop OEOs. This model assumes that the signal variation along thecavity is small, so that the order of the components does not affect the round-trip signal trans-mission. In previous work, we have developed a comprehensive model to study a single-loopOEO [6]. This model generalized the Yao-Maleki model and includes all of the physical effectsin the Yao-Maleki model as well as other physical effects that are needed to calculate importantfeatures of the OEO dynamics, such as the fast response time of the modulator, the ability of


the OEO to oscillate in several cavity modes, and amplitude fluctuations that are induced by theinput noise.

In this manuscript, we describe a computational model for single- and dual-injection-lockedOEOs (DIL-OEOs) that is a significant extension of the model that is described in Ref. [6],which only applied to a single-loop OEO. This extended model has been experimentally veri-fied [7, 8]. In order to obtain agreement between theory and experiment, we added to the modeldescribed in Ref. [6] a phase flicker (1/ f ) noise source, gain saturation in the RF amplifiers,and the spectrum of the RF filter. By extracting the magnitude of the phase flicker noise am-plitude from phase measurements in single-loop OEOs with different cavity lengths, we havefound that the phase flicker noise contains a strong component with an amplitude that dependson the square-root of cavity length. Therefore, in long-cavity OEOs (L > 5 km) the phase noiseat low frequencies ( f < 500 Hz) is dominated by the phase flicker noise. A good quantitativeagreement between theory and experiments is obtained for the phase noise spectrum and thespur levels of both the slave and the master loops in the DIL-OEO. In contrast to the reducedmodel that we previously described in [4], this model takes into account the full OEO dynamics,including the growth of the oscillator signal from noise, rather than assuming steady-state op-eration. Thus, unlike the model in [4], it is capable of determining the locking bandwidth of thetwo loops in the DIL-OEO. Moreover, this model takes into account the locations of the lumpedelements in each loop, rather than treating them as distributed elements. As a consequence, thismodel can accurately describe the effect of large lumped coupling between the loops of theDIL-OEO, as well as gain, loss, and gain saturation. Since this computational model is basedon the full physics, it is inherently more trustworthy than the reduced models. As is almost thecase when comparing simplified or reduced models to more complete or full models, there isa tradeoff between computational speed and accuracy. Our view is that for modeling OEOs,this tradeoff favors the use of the full model. It runs quickly — taking less than two minutes ofCPU time on a standard desktop computer for one set of parameters — and has allowed us toexamine a broad parameter range.

We used our model to theoretically study how to improve the DIL-OEO performance. Ourmodel predicted that it is possible to reduce the magnitude of the first spur that is obtained in themaster loop of the DIL-OEO by 20 dB relative to the level in the original experiments [7]. Thedecrease in the spur level is obtained by increasing the slave loop length by a factor of 10 andby using a strong injected signal with a power of −6 dB with respect to the oscillating signalin both loops. The increase in the slave loop length and the increase in the power injectionratio contribute about 10 dB each to the decrease in the spur level. Subsequently performedexperiments verified our model predictions, and guided by the theoretical results we were ableto significantly reduce the spur level. These experiments have been presented in part in [8] andwill be presented in full elsewhere.

The remainder of this paper is organized as follows. In Sec. 2 we review the single-loop OEOmodel, including the generation of the flicker noise, and we present the model for DIL-OEO. InSec. 3 we present comparison between the experimental results and the theoretical results forthe phase noise in a single-loop OEO and DIL-OEO. A good agreement is achieved betweentheory and experiment. In Sec. 4, we present a theoretical study of methods to reduce the spurlevel. The theoretical calculations that were subsequently verified experimentally show that asignificant reduction in the spur level can be achieved by increasing the loop delay of the slaveloop and optimizing the coupling between the loops. Finally, we summarize our results andconclude in Sec. 5.


Fig. 1. Schematic illustration of the single-loop OEO configuration

2. Model description

In this section we describe our model for the single-loop and dual-loop OEO configurations.As previously noted in [6], four different time/frequency scales play a role in the OEO. At thehighest frequency scale, we have the optical carrier in the optical fiber, which is close to 200THz. However, the role of the optical carrier is completely passive. The light merely serves tocarry the RF signal from one end of the optical fiber to the other and has no effect on the OEOdynamics. We therefore model the optical fiber as a fixed delay of the RF signal and ignore thedynamics on the optical frequency scale. At the next-highest frequency scale, we have the RFcarrier at around 10 GHz. At this scale, the RF filters in the OEO loops, which typically havea bandwidth of about 10 MHz, filter out harmonics of the RF signal that appear due to gainsaturation in the RF amplifiers and optical modulators. At the next-highest frequency scale, wehave the inverse of the round-trip time in the OEO loops. This time may be as short as 0.2 μsfor a 40 m loop, corresponding to a frequency of 5 MHz, and may be as long as 30 μs for a6 km loop, corresponding to 33 kHz. Finally, we have the frequency scale of the phase noise.We are interested in the frequency range from about 1 Hz to 100 kHz. A point that should beemphasized is that the last two scales are not well-separated. In most previous models [1, 4, 5],the behavior during one round-trip in an OEO loop was not resolved. As a consequence, theseprevious models are simpler than ours and use less CPU time than ours does; however, theyalso cannot reliably achieve the quantitative agreement with experiment that our model does.

By contrast, it is not necessary to resolve phenomena that occur on the time scale of the 100ps oscillation period of the RF carrier, since this time scale is widely separated from the twolonger time scales.

In a typical simulation of the DIL-OEO, we calculate the signal evolution for 0.01 seconds,corresponding to 50,000 round-trips in a short slave loop with a 40 m loop. This simulationtakes approximately 2 minutes on an IBM with a CPU speed of 2.33 GHz and 4 GB of RAM.


2.1. Modeling the oscillating signal and its noise in a single-loop OEO

In this sub-section, we review the single-loop OEO model; more details may be found in [6]. Aschematic illustration of the single-loop OEO is given in Fig. 1. Light from a laser is fed into anelectro-optic modulator, which is used to convert microwave oscillations into a modulation ofthe light intensity. The modulated light is sent through a long optical fiber and is then detectedusing a photodetector, which converts the modulated light signal into an electrical signal. Theelectrical signal is then amplified, filtered, and fed back into the electrical port of the modulator.

In the analysis of a single-loop OEO, it is assumed that due to the narrow bandwidth of theOEO filter, the voltage applied to the modulator, Vin(t), is approximately a sinusoidal wave withan angular carrier frequency ωc = 2π fc, a time-dependent phase φ(t), and a time-dependentamplitude |amod

in (t)|, so that

Vin(t) = |amodin (t)|cos[ωct −φ(t)] =

12

amodin (t)exp(−iωct)+ c.c., (1)

where amodin (t) = |amod

in (t)|exp[iφ(t)] is the complex envelope or the phasor of the voltage Vin(t).Since the OEO signal is narrow-band we assume that |dφ/dt|�ωc and d|amod

in |/dt �|amodin |ωc.

The phasor of the oscillating signal contains all the information in both the amplitude andphase noise spectrum. As noted previously, the nonlinear response of the OEO componentssuch as the electro-optic modulator and the RF amplifiers creates high-harmonic componentsthat are centered around angular frequencies mωc with m > 1, where m is an integer, but thesehigher harmonics are filtered out, so that it is sufficient to model only the propagation of thephasor that represents the signal around the angular carrier frequency ωc. The evolution of thephasor amod

in (t) in the OEO cavity is calculated by taking into account the effect of all the OEOcomponents: the electro-optic modulator, the fiber delay, the photodetector, the RF amplifierswith saturation, and the RF filter.

The electro-optic modulator that is analyzed in our model is a Mach-Zehnder modulator.We used the Jacobi-Anger expansion to calculate the first harmonic in the nonlinear modulatorresponse [1, 6]. Keeping only the first harmonic term, the phasor at the output of the photode-tector, aPD

out, is related to the phasor at the input RF port of the modulator, amodin , by

aPDout (T ) = −αP0ηρRcos(πVB/Vπ,DC)J1

(π

∣∣∣amodin (T )

∣∣∣/Vπ,AC

)exp [iφ (T )] . (2)

where α is the insertion loss in the modulator and the detector, P0 is the optical power at themodulator input, η is a parameter determined by the extinction ratio of the modulator (1 +η)/(1−η), ρ is the responsivity of the photodetector, R is the impedance at the output of thedetector, Vπ,DC and Vπ,AC are the modulator half-wave voltages for the DC and AC voltages,respectively, and VB is the DC bias voltage.

Our experimental setup [7] included three identical RF amplifiers, each with a small-signalgain of 20 dB. The gain saturation of the three cascaded RF amplifiers was measured and thenused in the model. The gain saturation curve is shown in Fig. 2(a); it describes the relationbetween output power of the 3 RF amplifiers Pout = |aamp

out (t)|2/2R and the input power Pin =|aamp

in (t)|2/2R, where aampin (t) = aPD

out(t).The model also takes into account the RF filter response and its dynamic effects. The dynamic

effects are taken into account by using the time history of the phasor, which is chosen to belonger than the response time of the filter. The filter implementation and its associated dynamiceffects are described in detail in [6]. The full width at half maximum (FWHM) of the RFfilter bandwidth that is used in our experimental setup [7] was equal to ΩF = 8 MHz. In ourmodel, we used the experimentally measured transmission function that is shown in Fig. 2(b).To ensure that the RF filter has a casual response, a linear phase chirp was added to the measuredtransmission function that corresponds to a delay of 0.1 μs.


We have implemented our model by discretizing the phasor of the oscillating signal alongthe loop using an array containing N points, which implies a time separation or resolution timeδ t = τ/N. The number of points was chosen so that the simulation bandwidth Δ f = 1/δ t isbroader than the RF filter FWHM bandwidth, ΩF. Furthermore, we checked that the results didnot change when we increased the number of points N.

Additive white Gaussian noise is included in the model and is added to the phasor of theoscillating signal. Additive white Gaussian noise is added at the output of the photodetectorand at the input of each of the RF amplifiers at each round trip. The spectral power densityof the additive noise at the output of the photodetector and at the input of the RF amplifiersis determined by evaluating the shot noise power density, ρSN, and the thermal noise powerdensity, ρth, respectively. The photodetector’s shot noise power density is evaluated using theformula: ρSN = 2eIPDR, where IPD = 〈|aPD

out(t)|〉τ/R is the photodetector’s current, averaged overone round trip, and 〈〉τ denotes averaging over the round-trip time. The spectral power densityof the thermal noise, given by ρth = (NF)kBT , is determined by the noise factor NF of the RFamplifiers. The noise factor was determined empirically, and in order to obtain the best matchbetween theory and experiment we typically used a noise factor of NF = 4. We added noiseto the oscillating phasor in the same manner that is described in [6]. During each round trip,we added N mutually independent noise variables wi, i = 1, ...,N, to the array of the oscillatingphasor, such that the variance of the noise variables is set by the relation 〈|wi|2〉τ/2R = ρ/τ ,where ρ is the noise power density. A complex Gaussian distribution is assumed, and each ofthe real and imaginary parts of the noise variables is normally distributed with a variance ρR/τ .The noise is added in the simulation after the photodetector and before each of the RF amplifierswith a noise power density of ρSN and ρth, respectively. We note that the main contribution ofthe thermal noise to the phase noise is from the noise that is added at the input of the first RFamplifier. As a result, the phase noise in our simulation is practically determined by the totalwhite noise that is added between the photodetector and the first RF amplifier, which has anoise power density of ρtotal = ρth +ρSN.

The experimental results indicate that phase flicker noise (1/ f ) is the dominant noise sourceat low frequencies ( f < 500 Hz) of long-cavity OEOs (L � 5 km). Therefore, we include inour model a phase flicker noise source. In electronic devices, such as RF amplifiers, flickernoise is generated at DC and then nonlinearly upconverted to the carrier frequency [10]. Thesame appears to be true in the optical domain where some combinations of fiber dispersionand nonlinearity, double-scattering processes, and environmental effects can convert electronicflicker noise to optical flicker noise and other odd powers of the frequency in the phase noise

−60 −40 −200

10

20(a)

Pin(dBm)

Pout(d

Bm

)

−20 0 20−100

−50

0(b)

f (MHz)

|F(f

)|2(d

B)

Fig. 2. (a) The measured electrical power at the output of the 3 RF amplifiers as a functionof the input power. (b) The measured spectrum of the RF filter as measured experimentallyand used in the model.


spectrum. Thus, the phase flicker noise is multiplicative. We model the phase flicker noise bymultiplying the phasor of the oscillating signal by exp[iθ(t)] at the output of the RF ampli-fiers, where θ(t) is a time-domain realization of flicker (1/ f ) noise, which has a power spectraldensity 〈Sθ ( f )〉 = b−1/ f rad2/Hz. We found that the effect of the phase noise was indepen-dent of the location in the loop where we added the flicker noise. The flicker coefficient b−1

is dimensionless, and it determines the power spectral density of the phase flicker noise. Theflicker coefficient was determined empirically in order to obtain the best match between theoryand experiment, and in a short-loop-length OEO we used a flicker coefficient of 1−2×10−12,which is a typical range for RF amplifiers [10]. Flicker noise is correlated, and it can be mod-eled by linearly filtering white Gaussian noise. Several different techniques exist. We used theapproach that is described in Ref. [11]. In sub-section 2.2, we describe in detail how the flickernoise was generated in our model.

During each round trip, we calculated the evolution of the phasor, taking into account theadditive white noise and the multiplicative flicker noise. It is necessary to record the phasoramod

in (t) over some large number NRT round trips in order to calculate the phase noise powerspectral density at low frequencies. The phase noise of the oscillating signal is calculated usingthe Fourier transform of the accumulated phasor [6]. The lowest frequency that can be resolvedis on the order of 1/Ttot where Ttot = NRTτ is the overall accumulation time of the phasor in thesimulation run. The power spectral density of the phase noise is then calculated by averaging thepower spectral density that we obtain from individual simulation runs over Navg realizations. Inour simulations, we used a range of accumulation times Ttot = 10−100 ms, a range of resolutiontimes δ t = 50− 60 ns, and a range of numbers of realizations Navg = 20− 100. We checkedthe convergence of our computational results by verifying that the power spectral density ofthe phase noise did not change when we increased the resolution time δ t or the number ofrealizations Navg.

2.2. Modeling the flicker noise

In our simulation model, we used the approach that is described in Ref. [11] to create a discretetime series, θk, k = 1, ...,M, with an averaged spectrum of 〈Sθ ( f )〉 = b−1/ f . The length of thetime series is determined by the ratio between the total accumulation time and the resolutiontime of the simulation, M = Ttot/δ t = NRTN. We started with discrete white Gaussian noisein the time domain, wk. The variance of the white noise was set so that 〈wk〉2 = 2πb−1. Thefiltering in the simulation was implemented in the frequency domain, so that

Θ(νn) = H(νn)W (νn), (3)

where n = 1, ...,M, νn = −1/2 +(n− 1)/M is the normalized fourier frequency, Θ(νn) is thefiltered noise, H(νn) is the filter response in the frequency domain given by

H(νn) = [1− exp(−2πiνn)]−1/2,H(νM/2+1) = 0,

(4)

and W (νn) is the discrete Fourier transform of wk defined by

W (νn) =M

∑k=1

wk exp[2πi(k−1)(n−1)/M]. (5)

The discrete time series of the flicker noise in the time domain is given by applying an inverseFourier transform to the filtered noise Θ(νn)

θk = (1/M)M

∑n=1

Θ(νn)exp[−2πi(k−1)(n−1)/M]. (6)


Fig. 3. Schematic description of the DIL-OEO. The DIL-OEO operates in a master-slaveconfiguration. The longer loop, referred to as the master loop, generates a harmonic signalwith a very low phase noise. The shorter loop, referred to as the slave loop, is used todecrease the amplitude of the spurs. Part of the master signal, Γ12, is injected into the slaveloop, as indicated schematically by the solid arrow. The dashed arrow indicates that part ofthe slave signal, Γ21, is coupled back into the master loop.

The discrete time series θk has an averaged spectrum of 〈Sθ ( f )〉 = b−1/ f . The series θk isgenerated at the beginning of each simulation run. During each round trip, we use N subsequentterms of the series to multiply the array of the phasor by exp(iθk), so that by the end of the runall the terms of the series are used only once.

2.3. Modeling dual-loop OEOs

Figure 3 shows a schematic description of the DIL-OEO configuration. The lumped compo-nents in the loop, as well as the added noise, are modeled in the same manner as was describedin sub-sections 2.1 and 2.2 for the single-loop OEO. The difference between the two configura-tions is clearly the presence of the couplers in the injection bridge which are used to injection-lock between the oscillating signals of the master-loop and the slave-loop in the DIL-OEO. Inthis sub-section we describe how the model takes into account the couplers and how it treatsthe time synchronization between the two oscillating signals.

We denote the loop delays τ1 and τ2, when τ2 ≤ τ1. Following the terminology in [2], werefer to the OEO loop with the longer loop delay, τ1, as the master loop, and to the OEO loopwith the shorter loop delay, τ2, as the slave loop. We assume an arbitrary carrier frequencyfc that is approximately equal to the expected oscillating frequency of the DIL-OEO, and wedenote the phasors in the master loop and the slave loop with respect to the carrier frequency asa1(t) and a2(t), respectively.

The coupling between the two loops is characterized by four complex coefficients, γi j, i, j =1,2: (

a′1(t)a′2(t)

)=

(γ11 γ21

γ12 γ22

)(a1(t)a2(t)

). (7)

where ai(t) and a′i(t) (i = 1,2) are the amplitudes before and after the coupling in the masterloop (i = 1) and in the slave loop (i = 2), respectively. The forward injection coefficient, Γ12 =|γ12|2, represents the relative injected power from the master to the slave loop and the backwardinjection coefficient, Γ21 = |γ21|2, represents the relative injected power from the slave to themaster loop. Unidirectional injection corresponds to Γ21 = 0.


The evolution in the DIL-OEO was simulated by calculating iteratively the change in thephasors a1(t) and a2(t) in a round trip in each of the OEO loops. In our model implementation,the phasors a1(t) and a2(t) were sampled with the same resolution time δ t, which is typically50 nanoseconds and was always chosen so that τ2/δ t = N2 is an exact integer. In a 40 m loopwe typically chose N2 = 4. In this case, we find τ1 = N1δ t + δτ1, where |δτ1| < δ t/2. In oursimulations, we retained N2 values of a2(t) and N1 values of a1(t) in two separate arrays. Thearrays of the phasor in the master loop and in the slave loop before the injection bridge aredenoted by a1(i1) and a2(i2), respectively, such that i1 = 1, ...,N1 and i2 = 1, ...,N2. The arraysof the phasor in the master loop and in the slave loop after the injection bridge are denotedby a′1(i1) and a′2(i2), respectively. The phasor elements were calculated in each OEO loop andwere coupled when they arrive at the coupler.

The time synchronization in the coupling between the two loops was implemented in themodel in the following manner: Let k = 1, ...,M be the iteration index and M be the number ofthe total accumulated terms in the simulation run in each of the loops, such that M = Ttot/δ t,and let us assume that δτ1 = 0. We let i1 = mod(i,N1)+ 1 and i2 = mod(i,N2)+ 1. Thus, thevariables i1 and i2 pass cyclically through the values i1 = 1, ...,N1 and i2 = 1, ...,N2. In eachiteration, we used a1(i1) and a2(i2), for which we calculated the phasors after the injectionbridge, a′1(i1) and a′2(i2), using Eq. (7). When i j = Nj, j = 1,2, we used the array of the phasorafter the bridge, a′j(i j), in order to calculate the array of the phasor before the bridge, aj(i j),for the following round-trip. The evolution of the phasor after the bridge in each loop wascalculated by taking into account the response of all the lumped components on the phasor, aswell as the additive white noise and the multiplicative flicker noise. The evolution of the phasorarray in each loop was calculated in the same manner as in the single-loop OEO model that wasdescribed in sub-section 2.1 and in [6].

We modeled the case δτ1 = 0 by adding a constant phase shift of δφ1 = −2π fcδτ1 to thephasor array of the master loop after it propagates for a time N1δ t that is approximately equal tothe round-trip duration of the master loop. Our model’s ability to treat the case δτ1 = 0 allowsus to take into account the incommensurability of the two loops, which is always present inpractice.

We note that the dual-loop OEO model includes all the dynamical effects that were studiedin the single-loop model, such as the cavity mode competition during the OEO start-up andtemporal amplitude oscillations. These dynamical effects and others are described in detail fora single-loop OEO in Ref. [6].

3. Comparison with experimental results

In this section we compare the theoretical and the experimental results for the phase noisein a single-loop OEO and in the DIL-OEO. Good agreement between the theoretical and theexperimental results is achieved for both the single-loop OEO and for the DIL-OEO. The Mach-Zehnder modulator’s measured parameters values are: Vπ,AC = 5 V, Vπ,DC = 3.15 V, VB = 2.6 V,and η = 0.7. We used the specified photodetector responsivity: 0.8 A/W at DC and 0.55 A/W at10 GHz. The measured optical power P0 is 17 mW and the impedance at the output of the pho-todetector was R = 50 Ω. The first step of the comparison was to compare the theoretical andthe measured RF power at the output of the detector. We added an effective loss between −0.4dB and −0.9 dB to the model in order to match the measured RF power. The photodetector’sshot noise power density, ρSN, was then determined from the round-trip averaged photodetec-tor’s current IPD. We empirically set the noise factor NF of the RF amplifiers — and hence thethermal noise power density ρth = (NF)kBT — and the flicker noise coefficient b−1, so that weobtained the best match between theory and experiment, as described in sub-section 3.1.


101

102

103

104

105

−150

−100

−50

f (Hz)Ph

ase

nois

e (d

Bc/

Hz)

Fig. 4. Comparison between the experimentally measured noise spectral density, SRF( f )(solid gray line) and the noise spectral density that is calculated by using a model thatexcludes flicker noise (dashed-dotted green line) and a generalized model that includesflicker noise (solid red line). All the theoretical curves were calculated assuming the samewhite noise, oscillation power, and small signal gain. The OEO loop-delay equals τ =31.7 μs and the phase flicker coefficient equals b−1 = 10−11.

3.1. Loop length dependance of the phase flicker noise in a single-loop OEO

The coefficient of the phase flicker noise and the noise factor of the RF amplifiers were de-termined empirically by comparing the theoretical and the experimental phase noise spectra.Figure 4 shows a comparison between theory and experiment in a single-loop OEO with alength of 6400 m, which corresponds to a loop-delay of τ = 31.7 μs. The figure shows that inorder to obtain good agreement between theory and experiment, a phase flicker noise sourcemust be added. Good agreement between theory and experiment was also obtained when theloop delay was equal to 0.5,2.7,7,10,15, or 27 μs, corresponding to loop lengthes between100 and 5400 m, when an appropriate amount of flicker noise was added. For all the OEO

0 2000 4000 60000

1

2

3x 10

−11

L (m)

b −

1

× 10−11

Fig. 5. Dependence of the phase flicker coefficient on the loop length as extracted fromsingle-loop OEO measurements (red dots). The dependence was found to be approximatelylinear (dashed line): b−1 = 2×10−12 +2.5×10−15L, where the fiber length L is measuredin meters . The accuracy of the extracted dots is limited by the accuracy of the measureddata, which is approximately 3 dB. Therefore, the error-bars of the extracted dots are equalto 3 dB.


lengths, the measured oscillation power at the photodiode was −22± 1 dBm and the oscil-lating power at the output of the RF amplifiers was equal to 23.3± 0.3 dBm. The total whitenoise power density that was used in the theoretical model for all the loop delays was equal toρN = 9×10−20±0.5×10−20 W/Hz, which can be obtained by assuming that the photodetectoris limited by shot noise and that the RF amplifiers have a noise factor NF = 4. The only freeparameters in the simulation that we changed when the cavity length was varied were the flickercoefficient b−1, which determines the power density of the phase flicker noise source, and theeffective loss. We varied the effective loss in the model between −0.4 dB and −0.9 dB in or-der to match the theoretical RF power of the photodetector output to the measured power. Wechose the flicker noise coefficient, b−1 so that the theoretical phase noise matches the measuredphase noise. Figure 5 demonstrates the dependence of the extracted flicker noise coefficient onthe cavity length. Each dot in Fig. 5 was extracted from the measured phase noise. The accu-racy of each dot is limited by the noise in the measured data, which approximately equals 3dB. The figure shows that the flicker coefficient can be divided into a noise component that isindependent of the cavity length b−1 = 2×10−12 and a component that depends on the cavitylength L. The component that does not depend on the loop length is consistent with the typicalflicker noise power that is observed in RF amplifiers [10]. There are several possible sourcesof the length-dependent flicker noise. These include: conversion of laser frequency noise intophase noise noise via dispersion and/or laser RIN noise into phase noise via the Kerr nonlinear-ity, double-scattering processes in the fiber such as double-Rayleigh backscattering or doubleBrillouin scattering, polarization mode dispersion, and environmental effects. This list is notexhaustive and we intend to study the exact causes of the length-dependent flicker noise in thefuture. The length-dependent flicker noise is the principal source of phase noise in the low fre-quency region ( f < 500 Hz) of long-cavity OEOs (L > 5 km). The dependance of the flickernoise power on the loop length that was obtained in the single-loop OEO was successfully usedas an input to our model for the DIL-OEO.

3.2. Comparison between theory and experiment in the DIL-OEO

In this sub-section we present a comparison between the theoretical and the measured phasenoise in a DIL-OEO [9]. The experimental setup is described in [7] and a schematic descriptionof the device is given in Fig. 3. A master loop with a length of 4196 m was coupled to a slaveloop with a length of 44 m. First, we compared the theoretical and the experimental results forboth the slave loop and the master loop in the free-running case when the two loops are notcoupled and function as single-loop OEOs. This case corresponds to setting Γ12 = Γ21 = 0 andΓ11 = −0.3 dB, Γ22 = −2.5. The other parameters used for the two loops were the same as asthose we used in the model for the single-loop OEO. We included white noise and flicker noisein the model. The power density of the white noise was equal to ρN,1 = 2.4×10−20 W/Hz forthe master loop and to ρN,2 = 9×10−21 W/Hz for the slave loop. The flicker noise coefficientthat we used was equal to b−1 = 10−11 for the master loop and b−1 = 10−12 for the slave loop.The flicker noise coefficient of the slave loop is consistent with the flicker noise coefficient thatwas measured for RF amplifiers [10]. However, the flicker coefficient in the master loop wasconsiderably higher than in the slave loop, which is in accordance with the results presentedin Fig. 5. We obtained excellent quantitative agreement between theory and experiments in theunlocked case, as can be seen in Fig. 6(a).

Figure 6(b) shows the phase noise in the slave and in the master loops when the two loopswere coupled and the injection power coefficients were Γ11 = −0.3 dB, Γ22 = −2.5 dB, andΓ12 = Γ21 = −20 dB. The coupling between the two loops was experimentally implementedusing a phase shifter before the coupling bridge in the same loop, so that the coupling co-efficients γi j are real numbers. We obtained good quantitative agreement between theory and


102

103

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f (Hz)

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e no

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(a) Free-running case

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e no

ise

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−130

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f (kHz)

Slave loop

Master loop

(b) Injection-locked case

Fig. 6. (a) Phase noise of the master loop (blue) and the slave loop (red) when the loopsare free-running and function as single-loop OEOs. (b) Phase noise of the master loop(magenta) and the slave loop (green) when the loops are injection-locked. The inset zoomsin on the first spur in the master and slave loops. Good agreement is achieved betweenthe experimental results (thin lines or light colors) and the theoretical results (thick linesor dark colors) when the loops are injection-locked. The injection power coefficients wereΓ11 =−0.3 dB, Γ22 =−2.5 dB, and Γ12 = Γ21 =−20 dB. Theory shows that the first spurin the master loop is about 20 dB lower than in the unlocked case.

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e no

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(a) Slave loop

40 50 60−150

−120

−90

f (kHz)

Phas

e no

ise

(dB

c/H

z)

Injection−lockedloop peaks

Free−runningloop peaks

(b) Master loop

Fig. 7. (a) Calculated phase noise for the free-running slave loop (red) and for the injection-locked slave loop (green) compared to experimental results (thin lines or light colors). Thephase noise within the locking range is determined by the master loop. (b) The calculatedfirst spur of the injection-locked master loop (magenta) compared to the spur in the free-running loop (blue). The spur is reduced by approximately 20 dB by injection-locking.The theoretical results (thick lines or dark colors) are also compared to the correspondingexperimental results (thin lines or light colors).


45 50 55

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(a)

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f (kHz)

(b)

−20 dB

0 dB

0 dB

−20 dB

−40 dB

−40 dB

Free running

Free running

Fig. 8. Calculated spectrum of the phase noise in the neighborhood of the first spur of themaster loop as RΓ varies for τ1 = 20 μs and (a) τ2 = 0.2 μs, (b) τ2 = 2 μs. We show resultsfor RΓ = −40 dB, −20 dB, and 0 dB. For comparison, we also show the spur level whenthe master loop is free-running and functions as a single-loop OEO.

experiment for the phase noise spectrum as well as for the spur levels in both OEO loops inthe injection-locked case as demonstrated in Fig. 6(b). Figure 7(a) shows that the model accu-rately describes the reduction of the phase noise in the slave loop within the frequency lockingrange of the two loops. Within the locking range, the phase noise in the slave loop is mainlydetermined by the phase noise of the master loop. Fig. 7(b) shows that the first spur level in theinjection-locked master loop is reduced by approximately 20 dB — from −95 dBc/Hz to −115dBc/Hz — compared to the spur level when it is free-running.

4. Theoretical study of approaches to decrease the first spur in the master loop

In this section, we describe our theoretical optimization of the OEO performance and our exper-imental verification of the theoretical predictions. A description of the experimental proceduresand results is summarized in [8] and will be described in full elsewhere. Our starting point forthe theoretical work was the experiments that are described in [7] and in sub-section 3.2. Weused the model presented in Sec. 2, and we varied the slave loop length and the power injectionratio, which is defined in the following paragraph. We found a reduction in the first spur byabout 20 dB in the injection-locked master loop, relative to the original experiments [7]. Thesecalculations were subsequently verified in experiments.

Our design goal is to reduce the spur level, while approximately maintaining the low phasenoise of the free-running master loop. We have chosen to focus on minimizing the phase noisein the master loop rather than in the slave loop since the phase noise of the slave loop is higheroutside the locking range.

We have found both theoretically and experimentally that we achieve the best results whenΓ12 � Γ21 and Γ11 � Γ22. We define the power injection ratio RΓ = Γ12/Γ11 = Γ21/Γ22, andwe will determine the optimum performance as we vary this quantity and the slave loop’s delaytime. In the computational work that we will present in this section, we have set Γ11 = Γ22 =0.25. In the experimental verifications that we present here, we set Γ11 to within 0.5 dB of Γ22.


−40 −20 0

−140

−120

−100

RΓ (dB)dB

c/H

z

τ2 = 0.2 μsτ2 = 2 μs

Fig. 9. Calculated dependence of the spur level on the power injection ratio when τ2 =0.2 μs (red triangles) and when τ2 = 2 μs (blue circles).

Figure 8 shows the calculated noise dependence of the phase noise around the first spur ofthe master loop as RΓ varies. In Fig. 8(a), we present results when the slave loop has a loopdelay of 0.2 μs, corresponding to a loop length of approximately 40 m, and in Fig. 8(b), wepresent results when the slave loop has a loop delay of 2.0 μs, corresponding to a loop lengthof approximately 400 m. We set RΓ = −40 dB, −20 dB, and 0 dB. The other parameters arethe same as those that we used for modeling the experimental setup that we described in theprevious section. For comparison, we also show the free-running case in both Fig. 8(a) andFig. 8(b). We recall that when the master loop is free-running, it functions as a single-loopOEO. In Fig. 9, we summarize the key results from Fig. 8 by showing the maximum spur levelas a function of RΓ. We find that the optimal power injection ratio in the original experimentalsetup [7], in which the slave loop delay is 0.2 μs, is equal to −20 dB. We also find that byincreasing the slave loop delay from 0.2 to 2.0 μs, the spur level in the master loop is reducedfrom −115 dBc/Hz to −125 dBc/Hz when RΓ = −20 dB and is further reduced to a level of−135 dBc/Hz by increasing RΓ to −6 dB. We note that the phase noise of the injection-lockedmaster loop in all the cases that we show in Fig. 9 is approximately equal to the phase noise ofthe free-running master loop. Thus, we theoretically predict a reduction of the first spur level inthe injection-locked master loop by approximately 20 dB relative to the spur level in the originalsetup, while maintaining approximately the same low phase noise. We note that although Fig. 9demonstrates the reduction of only the first spur level, other spurs are reduced as well. Thesecond spur level, for example, is reduced from −123 dBc/Hz in the original setup to less than−140 dBc/Hz by increasing the slave loop length from 0.2 μs to 2 μs and by increasing thepower injection ratio RΓ from −20 dB to −6 dB.

The first spur in the master loop can be further suppressed by increasing the slave loop’sloop delay beyond 2.0 μs. The level of the first spur in the master loop is mainly determined bythe phase noise of the free-running slave loop and the injection parameters. Thus, it is possibleto decrease the spur level in the master loop by increasing the length of the slave loop, whichdecreases the slave loop’s free-running phase noise. However, increasing the length of the slaveloop reduces its cavity mode spacing and can increase the level of the high-order spurs in themaster loop in cases where they nearly coincide with spurs in the slave loop. If these spurs liewithin the bandwidth of the RF filter, they can exceed the lower spurs in magnitude and degradethe performance of the DIL-OEO. We have not carried out a detailed study of this issue, as ithas not been present in our experiments to date, in which the maximum slave loop delay hasbeen 2.5 μs. However, we note that this issue will place a practical limit on how long it is


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f (kHz)Ph

ase

nois

e (d

Bc/

Hz)

Free running

τ2 = 0.2 μs,RΓ = −20 dB

τ2 = 2.5 μs,RΓ = −20 dB

τ2 = 2.5 μs,RΓ = −6 dB

Fig. 10. Experimentally measured spectrum of the phase noise in the vicinity of the firstspur. The spur level of the free-running master loop, which has a loop delay of τ1 = 20 μs,was −75 dBc/Hz (red). By injection-locking the master loop to a slave loop with a loopdelay of τ2 = 0.2 μs and a power injection ratio of RΓ = −20 dB, we reduced the firstspur to −110 dBc/Hz (green). Increasing the slave loop-delay to τ2 = 2.5 μs, and keepingthe same power injection ratio, we measured a spur of −109 dBc/Hz (cyan). The spur wasreduced to −129 dBc/Hz by increasing the power injection ratio to RΓ =−6 dB (magenta).

possible to make the slave loopThese predictions have been experimentally verified. As noted previously, we predicted op-

timal results when Γ12 � Γ21 and Γ11 � Γ22 — a point that we experimentally verified. Thebest spur reduction was achieved by increasing the slave loop-delay from 0.2 μs to 2.5 μs andby varying RΓ. The loop-delay of 2.5 μs is 0.5 μs longer than in the original theoretical stud-ies. The maximum reduction in the spur level was achieved by increasing RΓ from −20 dBto −6 dB. To achieve this large power injection ratio, we implemented a new bridge that hasΓ11 = −7.5 dB and Γ22 = −7.0 dB, so that this bridge has approximately 6 dB more loss thanin our earlier experiments [7]. We implemented this bridge by using four 3-dB couplers, withtwo placed in each loop. In each loop, the front coupler was connected to the rear coupler ineach loop and to the rear coupler in the other loop. With this configuration, we could obtainpower injection ratios, RΓ = Γ12/Γ11 � Γ21/Γ22 as large as 0 dB. We could then reduce RΓ byadding attenuators between the front coupler of one loop and the rear coupler of another loop.

We show the experimental results in Fig. 10. The first spur level of the free-running masterloop, which has a loop delay of 20 μs, was −75 dBc/Hz. By injection-locking the master loopto a slave loop with τ2 = 0.2 μs and RΓ = −20 dB, the spur level decreased to −110 dBc/Hz.When we increased the slave loop delay to τ2 = 2.5 μs and increased RΓ to −6 dB, the spurlevel further decreased to −129 dBc/Hz. These values are both within 2 dB of the calculatedresults with this set of parameters.

However, when we used a slave loop delay of τ2 = 2.5 μs with a power injection ratioRΓ =−20 dB in the experiments, we could not maintain a stable phase lock between the masterand the slave loops. The experimentally-measured spur level in this case was −109 dBc/Hz,instead of −120 dBc/Hz, as theoretically predicted. Our model assumes that the master andslave OEO loops are phase-locked, and it will not provide reliable answers unless there is agood lock. These results underline the importance of achieving a good phase lock — not onlyto achieve good agreement between theory and experiment, but also to obtain good performancefrom the OEO. We will say more about the conditions to achieve a good phase lock in a futurepublication.


Increasing the back-injection Γ21 from the slave loop to the master loop also reduces the firstspur, but at the expense of increasing the master phase noise within the locking range. Whenthe Q factor of the master loop, as defined in [1], is much higher than the Q factor of the slaveloop, Q1 Q2, the master phase noise in the injection-locked case is approximately unaffectedby the locking as long as Γ21 = Γ12 � 1 [4].

5. Conclusions

We have described a comprehensive computational model for studying single-loop and dual-loop OEOs. The model resolves the behavior of the signal during one round-trip in the OEO andtakes into account the lumped elements in the loop and both white and flicker noise sources.As a consequence, it allows us to reliably predict the spur level, the effect of large couplingbetween the two loops in the DIL-OEO, and other dynamical effects. In particular, it can be usedto accurately determine the variation of the phase noise in OEOs as their parameters change.An excellent agreement between theory and experiments was obtained for both the single-loopOEO and the DIL-OEO when its two loops are phase-locked.

We used the comparison between theory and experiment in a single-loop OEO to determinethe power spectral density of the white noise source and the phase flicker noise source. Wefound experimentally that the phase flicker noise power spectral density increases linearly withthe cavity length. Including this linear dependence in our model was necessary to achieve goodagreement between theory and experiment. The increase of the phase flicker noise as a functionof the cavity length is important since it limits the performance of long-cavity OEOs. Thephysical reasons for this dependence have yet to be determined.

We used the free parameters in our single-loop model — the linear loss as a function oflength, the power spectral densities of the white noise source and flicker phase noise source asa function of length — in our model of the DIL-OEO. Thus, our model of the DIL-OEO has noadditional empirical parameters. The model accurately predicts the phase noise in the masterand in the slave loops, the locking range in the slave spectrum, and the spur levels. The modelcan be used to study a general case of coupling between two OEO loops. In particular, it allowsus to reliably study OEO loops that are strongly coupled to each other.

Due to its accuracy and its ability to analyze OEOs with different configurations over a wideparameter range, our computational model can be used to optimize the performance of OEOs.We showed theoretically that it is possible to reduce the first spur in the master loop of the DIL-OEO by about 20 dB relative to our original experiments [7]. This reduction was subsequentlyverified experimentally [8]. We obtained this reduction by increasing both the loop-delay ofthe slave loop and by increasing the power injection ratio, so that the two loops are stronglycoupled.

Acknowledgement

This work is supported by DARPA MTO.


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